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Cookbook Review: When Those Great Easy Meals Aren't So Perfect On The Food Network
Jenn Garbee February 25, 2011
How many times are we going to say the magazine-cum-cookbook seems to be the publishing trend du jour? One more time, apparently, when the The Food Network Magazine releases Great Easy Meals: 250 Fun & Fast Recipes in the coming weeks. The press release promises the magazine's first cookbook "has all the features America loves about the bestselling magazine, and more." A cookbook that promises to be packed with glossy PR-pitches, perfectly styled recipes, and Food Network celebrity smiles? Oh goody.
Actually, this cookbook could actually be good. The television network, after all, is one of the few media outlets that can still boast a big test kitchen staff. And the book gets major brownie points for not being yet another platform for an individual Food Network celeb to tell us why everyone just loves their cooking so much. Instead, we are greeted in the Table of Contents by the pearly-white smiles of several celebs who promise to keep their soliloquies about their innermost kitchen lives to one-page via profiles throughout the book. But unfortunately, like the magazine, there is a glaring pitch-feel to those celebrity write-ups.
Guess What The Product Giveaway PR Was With Chiarello's Book?; Credit: flickr user chtk
Not to mention just plain silly soapbox comments. Consider the Neely's: "Probably the biggest misconception people have is that we don't cook because of all that cooking on TV," says Pat Neely. Really? We've never thought Food Network personalities actually cooked much of anything on the shows themselves. That's what the test kitchen/prep staff is for.
Fine, we should expect a Food Network cookbook to toot its own horn. But we're still having a really hard time justifying the book's "Food Network Kitchen's Secret Ingredients" glossary. Sure, it's perfectly fine to offer us photos and descriptions of rice vinegar and panko bread crumbs, even if arguably both have long since earned their pantry staple status in most home kitchens. But why must the photo of panko be a box of Progresso bread crumbs rather than some enticing shot of the bread crumbs themselves (there is no sidebar saying why Progresso is better than other brands)? It turns out that Food Network personality Michael Chiarello, who is featured in the cookbook, also happens to be a Progresso spokesperson. That chipotle photo? A front-and-center shot of Goya brand smoked jalapenos. Google "Food Network + Goya" and guess what comes up? Yup, a recent New York Times article describing the company's advertising expansion that includes a multi-million dollar Food Network campaign.
As this is a cookbook, perhaps we can fast forward the blatant advertorial and go straight to the recipes. After all, that's why we bought the book. Unfortunately, the recipe photos look like they spent entirely too much time getting their sesame seed makeup done. That most are shot on a clinical white background doesn't help, as it manages to make chicken enchiladas that should be oozing with tempting cheese, and what should be a fantastically messy chile relleno burger, look like they've just checked into the psych ward. That sort of perfection may work under the bright television lights, but in a cookbook it makes one wonder whether the food stylist forgot to bring along her Ritalin.
But we're trying to be objective, so we closed our eyes and pretended all of those clinical photos weren't there — on every single page. But the recipes still just aren't that interesting. Mini meat loaves, cheese-stuffed pork chops, black bean soup with turkey. The sort of dishes we've made in some form or another so many times, by now we already have a favorite recipe. Or we don't even need a recipe anymore.
Wait, might one argue this is the same sort of thing Chris Kimball and crew do over at America's Test Kitchen in similar books like The America's Test Kitchen Family Cookbook? The difference is the products ATK endorses are not backed with advertising dollars, and the ATK crew actually spends hundreds of hours perfecting that meatloaf — and afterward gives you way too much detail about how they came to some sort of tomato paste revelation. Not here. Great Easy Meals simply lists a recipe, no introductory note, and moves on to the next perfectly coiffed photo and basic recipe. Pretty formulaic. But hey, so are most Food Network celebrities.


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\section{Introduction}
Synchronization in networks of coupled oscillators is one of the most fundamental problems in networked dynamical systems. Networks of coupled ocillators present a rich dynamic behavior as reported in the vast literature on this topic \cite{SHS:00,FD-FB:13b,AJ-NM-MB:04}. Many real world phenomena can be modeled as a system of coupled oscillators: pacemaker cells in the heart \cite{michaels1987mechanisms}, neurons in the brain \cite{varela2001brainweb}, clock synchronization in computing networks \cite{baldoni2010coupling}, mobile sensor networks \cite{paley2007oscillator}, and power grids \cite{filatrella2008analysis,FD-FB:09z}. Considerable research in this field has been focused on studying the effect of network structure, coupling strengths, and nodal dynamics on various characteristics of synchronization including synchronizability, robustness, and adaptability \cite{di2007effects,sorrentino2007synchronizability,menck2014dead}.

Various metrics have been proposed in the literature to quantify and optimize the synchronization performance. A broad class of these metrics target the transient response, such as the ability of the network to resynchronize after perturbations \cite{donetti2005entangled,motter2005network,pecora1998master,kempton2015self}. In this context, synchronizability can be characterized by the required effort to synchronize the network \cite{sjodin2014price}, the speed of convergence to the synchronized manifold, or the range of coupling values for which a network with uniform coupling strength would synchronize \cite{pecora1998master}. Using the master stability framework proposed in the seminal paper \cite{pecora1998master}, it can be shown that the Laplacian algebraic connectivity and the eigenratio are two network-dependent measures that capture synchronizability. Using these metrics, numerous works have been reported on optimizing synchronizability \cite{pecora1998master,nishikawa2006maximum,donetti2005entangled,rad2008efficient,motter2005network,motter2013spontaneous,kempton2015self,skardal2015control,fardad2014design,clark2014global}. In particular, Nishikawa et al. \cite{nishikawa2006maximum} developed an extension of the master stability framework to the case of non-diagonalizable Laplacian matrices, and found a simple condition on the eigenvalues of the Laplacian that either maximizes the ratio between the maximum and minimum
stabilizing coupling strength, or minimizes the synchronization
cost (which has been defined as the sum of the coupling
strengths). In \cite{donetti2005entangled,rad2008efficient}, the authors proposed a collection of rewiring algorithms to enhance synchronizability. In the context of power grids, the authors in \cite{motter2013spontaneous} considered a power system with tunable parameters and used the maximal Lyapunov exponent to characterize the local stability of the swing equation in terms of these parameters. They used this characterization to specify the parameter values in order to enhance the synchronizability of the network.

In several real-world systems such as power grids, the individual dynamics and the coupling strengths are not uniform, and complete phase synchronization is compromised by nonidentical node dynamics. In \cite{FD-MC-FB:11v-pnas}, \emph{phase cohesiveness}, defined as the maximum steady-state phase difference across the edges of the network, was proposed as a condition of stable synchronization in networks of nonidentical Kuramoto oscillators. This metric quantifies partial phase synchronization and explicitly accounts for the simultaneous effect of the network topology, the coupling strengths, and the individual dynamics on the local stability of the synchronous solution. Therefore, we use phase cohesiveness as our design criterion in order to develop an optimization framework for designing the parameters of the network. For the exposition, we use the widely adopted model of Kuramoto oscillators which governs the dynamics of power grids \cite{filatrella2008analysis}. More specifically, we address the following two design problems:
\begin{enumerate}[leftmargin=*]
	\item  \emph{Design of natural frequencies}: In this problem, we assume that the network structure and the coupling strengths are given and the natural frequencies of the oscillators are the design variables. The network designer is able to tune the natural frequency of each oscillator by incurring a cost. The objective to minimize the tuning cost and guarantee a desired level of phase cohesiveness.

	\item \emph{Design of link weights}: In this second problem, we assume that the natural frequencies of the oscillators are given and the link weights are the design variables. We assume that the network designer is able to tune the weights of each edge by incurring a cost. The objective of the designer is then to design the edge weights in order to minimize the tuning cost, while guaranteeing a desired level of phase cohesiveness. To be consistent with practical aspects, we will also address the case in which the natural frequencies of the oscillators are uncertain with known bounds.
\end{enumerate}

The framework herein proposed can be used in a wide range of practical applications. In power grids, cascading failures are initiated due to the loss of generation power, or overloading of transmission lines. The common remedial actions to contain a cascade are cost-effective redispatching of generation, load shedding (i.e., tuning the natural frequencies in our framework), or switching on and off new lines \cite{linnemann2011modeling,barrows2012transmission}.
In another power grid example, new generators and loads are required to be connected to an existing grid, and finding an optimal structure within the new facilities that connect them to the rest of the grid is of interest.
Another application can be found in the context of clock synchronization in networks of distributed processors, where the natural frequencies of the processors are subject to uncertainty as a result of manufacturing defects. Therefore, the communication network connecting the processors must be designed to achieve synchronization with a pre-specified phase cohesiveness, while ensuring robustness against uncertainties in the natural frequencies \cite{simeone2008distributed}. Other applications include sparsity promoting network design \cite{siami2015network}, and the analysis of the well-known Braess' paradox (i.e., the counter-intuitive phenomenon of losing synchronization as the result of adding new edges \cite{witthaut2012braess}). We will provide numerical examples of these applications in Section \ref{section:NUMERICAL SIMULATIONS}.

The rest of the paper is organized as follows. In Section \ref{section:BACKGROUND AND PROBLEM STATEMENT}, we provide some preliminaries and formally state the network design problems. In Section \ref{section:OPTIMAL COHESIVENESS}, we develop a convex, tractable framework to solve the network design problems under consideration. Illustrative examples are highlighted in Section \ref{section:NUMERICAL SIMULATIONS}. Concluding remarks are drawn in Section \ref{section:CONCLUSIONS}.

\section{Problem statement}
\label{section:BACKGROUND AND PROBLEM STATEMENT}

\subsection{\label{sub:Graph-Theory}Preliminaries and notation}

\textit{Notation}: Let $\mathbb{R}$, $\mathbb{R}_{+}$, and $\mathbb{R}_{++}$ be the set of real, nonnegative, and strictly positive numbers. Let $\mathbf{1}_n$
and $\mathbf{0}_n$ be the $n$-dimensional vector of unit and zero entries.
The set $\{1,\ldots,n\}$ is denoted by $[n]$. Given an $n$-tuple $(x_1, . . . , x_n)$, let $\mathbf{x} \in \mathbb{R}^n$ be the associated column vector. The infinity norm of $\mathbf{x}$ is denoted as $\|\mathbf{x}\|_{\infty}=\max_i |x_i|$, the $\ell_1$ norm as $\|\mathbf{x}\|_1=\sum_{i=1}^{n} |x_i|$, and the $\ell_0$ norm as $\|\mathbf{x}\|_0=\mbox{card}(\{i\in[n] \colon x_{i} \neq 0 \})$, where $\mbox{card}(\cdot)$ denotes the cardinality of a set. For vectors $\mathbf{x}$ and $\mathbf{y}$, the inequality $\mathbf{x} \leq \mathbf{y}$ is component-wise. We define the vector-valued function $\mathbf{sin}(\mathbf{x}) = (\sin(x_1),\cdots,\sin(x_n))^\top$.
We denote by $\mathbb{S}^{n\times n}$ the set of $n\times n$ real, symmetric matrices. For square matrices $A$ and $B$, we write $A \succeq B$ if and only if $A-B$ is positive semidefinite.

\textit{Elements of spectral graph theory}: A graph is defined as $G=\left(\mathcal{V},\mathcal{E}\right)$,
where $\mathcal{V}$ is a set of $n$ nodes (or vertices) and $\mathcal{E}$
is a set of $m$ undirected edges (or links). We assume that the graph
is connected and has no self-loops. We consider graphs with weights
associated to both edges and nodes. We denote the weight of an edge $e=\left\{ i,j\right\} \in\mathcal{E}$
as $w_{e}=w_{ij}$. The \emph{weighted adjacency matrix} of a undirected graph $G$, denoted by $A_{G}=\left[a_{ij}\right]$,
is an $n\times n$ symmetric matrix defined entry-wise as $a_{ij}=w_{ij}$
if $\left\{ i,j\right\} \in\mathcal{E}$, and $a_{ij}=0$, otherwise.
The \emph{weighted Laplacian matrix} of $G$ is defined as $L=\mbox{diag}\left(A_G \mathbf{1}_n\right)-A_{G}$. The (unweighted) \emph{incidence matrix} of $G$ is defined for a directed and labeled version of the undirected graph $G$, as follows. First, we label each edge in the graph with a unique label $e\in\left\{ 1,\ldots,m\right\} $
and assign an arbitrary direction to it. In other words, we substitute
each undirected edge $\left\{ i,j\right\} \in\mathcal{E}$ for an
ordered pair $\left(i,j\right)$, in arbitrary order. For the ordered
pair $\left(i,j\right)$, we say that $i$ is the source and $j$
is the sink of the directed edge. For the resulting directed graph,
the incidence matrix $B=\left[b_{ie}\right]\in\mathbb{R}^{n\times m}$
is defined component-wise as $b_{je}=1$ if $j$ is the sink node
of edge $e$, $b_{ie}=-1$ if $i$ is the source node of edge $e$,
and $b_{ke}=0$ otherwise. For $\mathbf{x}\in\mathbb{R}^{n}$, notice
that $\left(B^{\top}\mathbf{x}\right)_{e}=x_{j}-x_{i}$ for a link
$\left(i,j\right)$ labeled $e$. For a weighted graph, we define the edge-weight vector $\mathbf{w}=\left(w_{1},\cdots,w_{m}\right)^{\top}$,
where $w_{e}$ is the weight of the edge labeled $e$. The Laplacian matrix of the weighted graph can be written as $L=B\mbox{diag}\left(\mathbf{w}\right)B^{\top}$ \cite{biggs1993algebraic}. \ifx Therefore, the Laplacian of a graph with nonnegative weights is positive semidefinite. In fact, the smallest eigenvalue of the Laplacian is always zero. \fi If the graph is connected, we have
that $\ker\left(B^{\top}\right)=\ker\left(L\right)=\mbox{span}\left({\mathbf{1}_{n}}\right)$,
and the second smallest eigenvalue of the Laplacian, called the \emph{algebraic connectivity},
is strictly positive \cite{biggs1993algebraic}. The \emph{Moore-Penrose pseudoinverse} of the Laplacian is defined as $L^{\dagger}=\left(L+\frac{1}{n}\mathbf{1}_n\mathbf{1}_n^{\top}\right)^{-1}-{\frac{1}{n} \mathbf{1}_n\mathbf{1}_n^{\top}}$. \ifx Equivalently, if the eigenvalue decomposition of $L$ is given by $L=U\mbox{diag}(\{0,\lambda_2,...,\lambda_n\})U^\top $, its Moore-Penrose pseudo inverse is given by $L^{\dagger}=U\mbox{diag}(\{0,\lambda_2^{-1},...,\lambda_n^{-1}\})U^\top $ \cite{biggs1993algebraic}.\fi For any connected graph with $n$ vertices, the identity $LL^{\dagger}=L^{\dagger}L=I_n-\tfrac{1}{n} \mathbf{1}_n\mathbf{1}_n^{\top}$ holds.

\subsection{\label{sub:Synchronization-in-Networks}Synchronization in networks of heterogeneous oscillators}
\ifx In this section, we describe the dynamics of the network of coupled
oscillators under study. \fi Consider a set of $n$ Kuramoto oscillators coupled via a connected weighted, undirected graph $G$. The temporal evolution of the network satisfies the following set of nonlinear ODE's,
\begin{align}  \label{eq:Kuramoto model}
\dot{\theta}_{i}(t)=\omega_{i} - \sum_{j=1}^{n}a_{ij}\sin\left(\theta_{i}(t)-\theta_{j}(t)\right),\forall i\in\mathcal{V},
\end{align}
where $\theta_{i}(t)\in\mathbb{R}$ is the angular position of the
$i$-th Kuramoto oscillator, $\omega_{i}$ is its natural frequency, and $a_{ij}\geq0$
is the $(ij)$-th entry of the weighted adjacency matrix of $G$. This dynamics can be written in matrix form as
\begin{equation} \label{eq: Kuramoto model-matrix}
\dot{\boldsymbol{{\theta}}}(t)=\mathbf{f}(\boldsymbol{\theta}(t)) = \boldsymbol{\omega}-BW \mathbf{sin}(B^\top \boldsymbol{\theta}(t)),
\end{equation}
where $\boldsymbol{\theta}(t)=(\theta_1(t),\cdots,\theta_n(t))^\top $,  $\boldsymbol{\omega}=(\omega_1,\cdots,\omega_n)^\top $, $B$ is the incidence matrix of $G$, $W=\mbox{diag}\left(\mathbf{w}\right)$ is the diagonal matrix of edge weights, and $\mathbf{w}=(w_1,\ldots,w_m)^\top$ where $w_e$ is the weight of the $e$-th edge in the graph. For further reference, we denote the Kuramoto model in \eqref{eq: Kuramoto model-matrix} by $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$. The following definition characterizes the notion of synchronization for $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$.

\begin{definition} \label{def: frequency-synchronization}
	A solution to the coupled oscillator model (\ref{eq:Kuramoto model}) is said to be \emph{frequency-synchronized} if $\lim\limits_{t \to \infty} |\dot{\theta}_i(t)-\dot{\theta}_j(t)|=0 ,\ \forall \{i,j\} \in \mathcal{E}$. Similarly, a frequency-synchronized solution to the coupled oscillator model (\ref{eq:Kuramoto model}) is said to be \emph{phase-synchronized} if $\lim\limits_{t \to \infty} |\theta_i(t)-\theta_j(t)| (\emph{mod} \ 2\pi)=0, \forall \{i,j\}\in \mathcal{E}$.
\end{definition}

Phase synchronization can only be achieved if all the natural frequencies are identical; otherwise, one could only achieve frequency synchronization. For a frequency-synchronized solution, the angular frequencies of the oscillators converge towards a common asymptotic frequency given by the average of the natural frequencies, i.e., $ \omega_{s} = \tfrac{1}{n}\sum_{i=1}^{n}\omega_{i}$, \cite{FD-MC-FB:11v-pnas}. Thus, the frequency-synchronized solution satisfies $\lim\limits_{t \to \infty} (\mathbf{\boldsymbol{\theta}}(t)-\mathbf{\boldsymbol{\theta}}_s(t))(\mbox{mod}\ 2\pi) =\mathbf{0}_n$, where
\begin{align} \label{eq: synchronized solution}
\mathbf{\boldsymbol{\theta}}_s(t)=(\omega_s t) \mathbf{1}_n+\mathbf{\boldsymbol{\theta}}^{\star},
\end{align}
for some $\mathbf{\boldsymbol{\theta}}^{\star} \in \mathbb{R}^n$ such that $\dot{\mathbf{\boldsymbol{\theta}}}_s(t)=\mathbf{f}(\mathbf{\boldsymbol{\theta}}_s(t))$.  It follows from \eqref{eq: synchronized solution} and Definition \ref{def: frequency-synchronization} that in frequency synchronization, we have that $\lim\limits_{t \to \infty} |\theta_i(t)-\theta_j(t)| = |\theta_i^{\star}-\theta_j^{\star}|, \forall \{i,j\}\in \mathcal{E}$.

\begin{definition}
	For any frequency-synchronized solution $\mathbf{\boldsymbol{\theta}}(t)$ of the Kuramoto model $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$, the corresponding \emph{phase cohesiveness} is defined as
	\begin{align}
	\varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})= \underset{ \{i,j\} \in \mathcal{E}}{\max} \lim\limits_{t \to \infty} |\theta_i(t)-\theta_j(t)| (\emph{mod} \ 2\pi).\nonumber
	\end{align}
\end{definition}
The phase cohesiveness can also be written as
\begin{align} \label{eq: phase cohesiveness expression}
\varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})&=\underset{ \{i,j\} \in \mathcal{E}}{\max} |\theta_i^\star-\theta_j^\star| &(\mbox{mod} \ 2\pi) \nonumber\\
&=\|B^\top \mathbf{\boldsymbol{\theta}}^{\star}\|_{\infty} &(\mbox{mod} \ 2\pi)
\end{align}
In order to further characterize the phase cohesiveness, we study the relationship between $\mathbf{\boldsymbol{\theta}}^{\star}$ and the network parameters $B,\mathbf{w}$, and $\mathbf{\boldsymbol{\omega}}$ in the following subsection.

\subsection{Fixed points and stability}\label{Subsec: Fixed Point}
We now study the synchronized solution given by \eqref{eq: synchronized solution}. Without loss of generality, we can assume that $\omega_s=\mathbf{\boldsymbol{\omega}}^\top \mathbf{1}_n=0$. Under this assumption, we have that $\mathbf{\boldsymbol{\theta}}_s(t)=\mathbf{\boldsymbol{\theta}}^{\star}$. It follows that $\mathbf{0}_n=\dot{\mathbf{\boldsymbol{\theta}}}_s=\mathbf{f}(\mathbf{\boldsymbol{\theta}}_s(t))=\mathbf{f}(\mathbf{\boldsymbol{\theta}}^{\star})$, i.e., the frequency-synchronized solution satisfies the fixed-point of (\ref{eq: Kuramoto model-matrix}),
\begin{align}
\boldsymbol{\omega}-BW\mathbf{sin}(B^\top \boldsymbol{\theta}^{\star})=\mathbf{0}_n. \label{eq: fixed point}
\end{align}
The following lemma will be used to characterize $\mathbf{\boldsymbol{\theta}}^{\star}$, and is a generalization of the result in \cite{FD-MC-FB:11v-pnas}.
\begin{lemma} \label{lemma: fixed point}
	Denote $F \in \mathbb{R}^{m\times{(m-n+1)}}$ as the matrix whose columns span the null space of $B$ (i.e., $BF=0$). Then, for any arbitrary $r \in \mathbb{R}$,  the fixed points of \eqref{eq: Kuramoto model-matrix} satisfy the following equation
	\begin{equation} \label{eq: complete solution}
	\mathbf{sin}(B^\top \boldsymbol{\theta}^{\star})=W^{r-1}B^\top (BW^{r}B^\top )^{\dagger}\boldsymbol{\omega} + W^{-1}F\mathbf{y},
	\end{equation}
	for some proper vector $\mathbf{y} \in \mathbb{R}^{(m-n+1)}$ that satisfies
	\begin{align} \label{eq: y vector}
	&\|W^{r-1}B^\top (BW^{r}B^\top )^{\dagger}\boldsymbol{\omega} + W^{-1}F\mathbf{y}\|_{\infty} \leq 1, \\
	&F^\top \boldsymbol{\mathbf{sin}}^{-1}\left(W^{r-1}B^\top (BW^{r}B^\top )^{\dagger}\boldsymbol{\omega} + W^{-1}F\mathbf{y}\right)=\mathbf{0}_{m}.\nonumber
	\end{align}
\end{lemma}
	\begin{pf}
		In Appendix \ref{appendix_lemma_fixed_point}.
	\end{pf}

The general result \eqref{eq: complete solution} includes special cases for different values of $r$. For instance, when $r=0$, we recover the synchronization solution obtained by a primal-dual optimization approach ( see \cite{zhou2015new} Eq. (22) ) , and $r=1$ corresponds to the synchronization condition reported in \cite{FD-MC-FB:11v-pnas}.  The next lemma, proved in \cite{FD-MC-FB:11v-pnas}, provides a sufficient condition for the stability of the fixed points satisfying \eqref{eq: fixed point}.
\begin{lemma} \label{lemma: stability of fixed points}
	Any fixed point of \eqref{eq: Kuramoto model-matrix} is locally exponentially stable if $\|B^\top \boldsymbol{\theta}^{\star}\|_{\infty} < \tfrac{\pi}{2}$.
\end{lemma}

It follows from the last lemma and \eqref{eq: phase cohesiveness expression} that the condition $\varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}}) < \pi/2$ is sufficient for local exponential stability of the fixed point. As a special case, in trees with uniform coupling weights, $F\mathbf{y}=\mathbf{0}_{m}$ and $\mathbf{w}=k\mathbf{1}_{n-1}$ where $k>0$ is the coupling strength, the phase cohesiveness reduces to $\sin(\varphi(B,k\mathbf{1}_{n-1},\mathbf{\boldsymbol{\omega}}))=k^{-1}{B^T(BB^T) ^{\dagger}\boldsymbol{\omega}}$. Consequently the stability condition in Lemma \ref{lemma: stability of fixed points} translates into the condition $k>\|B^T(BB^T) ^{\dagger}\boldsymbol{\omega}\|_{\infty}$ for tree graphs.

Notice that $\boldsymbol{\theta}^{\star}$ in Lemma \ref{lemma: fixed point} is characterized in terms of the implicit equations \eqref{eq: complete solution} and (\ref{eq: y vector}). These equations are hard to solve for arbitrary topologies. In fact, it was shown in \cite{taylor2015finding} that even finding the number of nonzero stable fixed points over the full space of phase angles $[0,2\pi)$ is NP-hard. Nevertheless, the following synchronization criterion was proposed in \cite{FD-MC-FB:11v-pnas}.
\begin{criterion} \label{sync criterion} The Kuramoto model $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$ has a unique and stable frequency-synchronized solution $\mathbf{\boldsymbol{\theta}}^{\star}$ such that $|\theta_i^{\star}-\theta_j^{\star}| \leq \gamma < \pi/2$ for every $\{i,j\} \in \mathcal{E}$ if
	\begin{equation}
	\left\Vert B^{\top}L^{\dagger}\boldsymbol{\omega}\right\Vert _{\infty}\leq\sin\left(\gamma\right).\label{eq:sync condition}
	\end{equation}
	where $L=B\emph{diag}(\mathbf{w})B^\top$.
\end{criterion}
The above criterion implies that when $\|B^\top L^{\dagger} \mathbf{\boldsymbol{\omega}}\|_{\infty}<1$, the phase cohesiveness satisfies
\begin{align} \label{eq: criterion equivalent}
\quad \varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}}) \leq \sin^{-1}(\| B^{\top}L^{\dagger}\boldsymbol{\omega}\| _{\infty}).
\end{align}
In other words, \eqref{eq: criterion equivalent} provides us with a tight upper bound for phase cohesiveness in terms of $B$, $\mathbf{w}$, and $\mathbf{\boldsymbol{\omega}}$. In particular, the condition $\|B^\top L^{\dagger}\mathbf{\boldsymbol{\omega}}\|_{\infty}<1$ guarantees local stability, according to lemma \ref{lemma: stability of fixed points}. \ifx Notice also that the upper bound $\|B^\top L^\dagger\mathbf{\boldsymbol{\omega}}\|_{\infty}$ is a particular solution of \eqref{eq: complete solution} with $r=1$ and $F=\mathbf{0}_{m\times (m-n+1)}$.\fi

\begin{remark}
	It was shown in \cite{FD-MC-FB:11v-pnas} that the criterion \ref{sync criterion} is provably correct for various
	network topologies and weights including the extremal cases of the
	sparsest (acyclic) and densest (complete) graphs.
	By statistical means, it has also been shown that the inequality (\ref{eq:sync condition})
	is extremely accurate for a broad set of random network topologies
	and weights, as well as for various standard power network test cases
	\cite{FD-MC-FB:11v-pnas}.
\end{remark}

\ifx
\begin{remark} \label{cor: tree graphs}
	For tree graphs, the null space of $B$ is trivial, i.e., $F=\mathbf{0}_{m \times (m-n+1)}$; hence, according to \eqref{eq: complete solution} the fixed point solution becomes independent of $r$. More specifically, using $r=1$, we obtain $\mathbf{sin}(B^\top\boldsymbol{\theta}^{\star})=B^\top L^{\dagger}\boldsymbol{\omega}$ and therefore, the phase cohesiveness bound in \eqref{eq: criterion equivalent} is tight for acyclic graphs. On the other hand, for $r=0$, we get $\mathbf{sin}(B^\top\boldsymbol{\theta}^{\star})=W^{-1}B^\top (BB^\top)^{\dagger}\boldsymbol{\omega}$
\end{remark}
\fi
In the next section, we use the upper bound in \eqref{eq: criterion equivalent} to develop the optimization framework for designing the natural frequencies of the network.

\ifx
\subsection{\label{sub:sync-metric}Synchronization metrics}
According to Criterion \ref{criterion}, the infinity norm of the approximate solution \eqref{eq: approximate solution} provides an upper bound for the cohesiveness of the synchronized phases. This upper-bound is tight for most real world oscillators. Therefore, we construct the metrics of synchronization based on this upper bound.
\medskip

\begin{enumerate}[(a)]

\item \textbf{Phase cohesiveness}: quantifies an upper-bound for the maximum phase difference across the edges. The infinity norm of \eqref{eq: sin(phistar)} yields the phase cohesiveness $\Phi: \mathbb{R}^m_{+} \times \mathbb{R}^n \to \mathbb{R}_{+}$ as follows:
	\begin{align} \label{eq: phase cohesiveness}
	\Phi(\mathbf{w},\boldsymbol{\omega}) &=\|B^\top (BWB^\top )^{\dagger}\boldsymbol{\omega}\|_{\infty},
	\end{align}
	The following definition is the consequence of relating lemma \ref{lemma: stability of fixed points} to Criterion \ref{criterion}.
\begin{definition}
	A network of oscillators with incidence matrix $B$, weight vector $\mathbf{w}$ and vector of natural frequency $\mathbf{\boldsymbol{\omega}}$ is synchronizable if $\Phi(\mathbf{w},\mathbf{\boldsymbol{\omega}}) < 1$.
\end{definition}
\begin{remark}
	Proof of Lemma \ref{lemma: stability of fixed points} implies that as $\Phi(\mathbf{w},\boldsymbol{\omega}) \to 1$, the second largest eigenvalue of the Jacobian in increases toward the origin, implying that phase cohesiveness captures the margin of stability for the synchronized solution.
\end{remark}

\begin{remark}
	For particular graph structures, the critical coupling \eqref{eq: Critical Coupling} can be simplified. In particular, for chain and star graphs, the critical coupling can be rewritten as
	\begin{align}
	&\Phi^{chain}(\mathbf{w},\boldsymbol{\omega})=\underset{1 \leq j \leq n-1}{\max}\left|\sum\limits_{i=1}^{j}\dfrac{(\omega_i-\omega_{s})}{w_i}\right|, \\  &\Phi^{star}(\mathbf{w},\boldsymbol{\omega})=\underset{2 \leq j \leq n}{\max}\left|\dfrac{\omega_i-\omega_{s}}{w_i}\right|, \nonumber
	\end{align}
	Dekker et al.\cite{dekker2013synchronization} have studied the approximation of the expected critical coupling for various classes of tree when frequencies are uniformly distributed on the unit interval.
\end{remark}

\item \textbf{Phase efficiency} quantifies the collective  performance of the whole network from an energy perspective. The phase efficiency is quantified by the weighted sum of squares of the phase differences:
\begin{align} \label{eq: phase efficiency}
\mathcal{P}(\mathbf{w},\boldsymbol{\omega})&=\mathbf{sin}^\top (\boldsymbol{\varphi}^{\star}) Q\mathbf{sin}(\boldsymbol{\varphi}^{\star}) \nonumber \\
&=\|B^\top (BWB^\top )^{\dagger}\boldsymbol{\omega}\|^{2}_{2,Q}
\end{align}
\begin{rem}
	In power networks, it can be shown that the instantaneous resistive power loss is proportional to the square of phase differences weighted by the conductances of the transmission lines \cite{sjodin2014price}.
\end{rem}

\ifx
\begin{rem}
	For the case of uniform natural frequencies, we have that $\boldsymbol{\omega}=\alpha \mathbf{1}_n$ and since $(BW^rB^\top )^{\dagger}\mathbf{1}_n=0$, it follows from \eqref{eq: phase cohesiveness}, \eqref{eq: phase efficiency}  all performance measures approach zero as $\mathbf{\boldsymbol{\omega}} \to \alpha
	\mathbf{1}_n$, meaning, the network perfectly synchronizes. In this case, the coupling weights only determine the speed of synchronization.
\end{rem}
\fi

\end{enumerate}

\fi
\section{Natural frequency design problem}
\label{section:OPTIMAL COHESIVENESS}

Consider a Kuramoto model $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$ with a given network structure ($B$) and link weights ($\mathbf{w}$). In the frequency design problem, our objective is to design the natural frequencies of the oscillators ($\mathbf{\boldsymbol{\omega}}$) to achieve a desired level of phase cohesiveness $\gamma_d \in [0, \pi/2)$ at a minimum design cost. In our setup, we assume that the network designer is able to tune the natural frequencies by incurring a cost described by a convex function. This problem can be mathematically stated as follows.
 \begin{problem}\textbf{\emph{(Cohesiveness-constrained frequency design)}}
	\label{prob:Optimal Budget Problem-Frequency design} Assume we are given the following elements: (i) a connected undirected network with incidence matrix $B$, (ii)
	a nonnegative vector of link strengths $\mathbf{w}_{0}\in\mathbb{R}_{+}^{m}$,
	(iii) a convex frequency-tuning cost function $g_{\mathcal{V}}\left(\mathbf{\boldsymbol{\omega}}\right): \mathbb{R}^n \to \mathbb{R}$, (iv) a closed convex feasible design set $F_{\mathbf{\boldsymbol{\omega}}}\subset\mathbb{R}^{n}$, (v) a desired level of phase cohesiveness $\gamma_d \in [0,\pi/2)$, and (vi) a synchronizing frequency $\omega_s \in \mathbb{R}$. Find the optimal vector of natural frequencies, denoted by $\mathbf{\boldsymbol{\omega}}^{\star}$, that solves the following optimization problem:
	%
	\begin{align} \label{eq: Cohesiveness-Constrained Frequency Design}
	\mathbf{\boldsymbol{\omega}}^{\star}=\arg \underset{\boldsymbol{\omega}\in F_{\mathbf{\boldsymbol{\omega}}}}{\min} & \quad  g_{\mathcal{V}}\left(\mathbf{\boldsymbol{\omega}}\right)\\
	\emph{s.t. } & \quad  \|B^\top L^{\dagger}\mathbf{\boldsymbol{\omega}}\|_{\infty}\leq \sin(\gamma_d), \nonumber \\
	& \quad \frac{1}{n}\boldsymbol{1}_n^\top   \boldsymbol{\omega}=\omega_s, \nonumber
 	\end{align}
	%
	where $L=B\emph{diag}(\mathbf{w}_0)B^\top $.

	Then, by Criterion \ref{sync criterion}, the phase cohesiveness of the resulting network $\mathcal{K}(B,\mathbf{w}_0,\mathbf{\boldsymbol{\omega}}^{\star})$ would satisfy
	$\varphi(B,\mathbf{w}_0,\mathbf{\boldsymbol{\omega}}^{\star}) \leq \gamma_d$ with minimum tuning cost given by $g_{\mathcal{V}}(\mathbf{\boldsymbol{\omega}}^{\star})$.
\end{problem}

Problem \ref{prob:Optimal Budget Problem-Frequency design} is a convex optimization problem, since the function $\|B^\top L^{\dagger}\mathbf{\boldsymbol{\omega}}\|_{\infty}$ is the point-wise maximum of linear functions of $\mathbf{\boldsymbol{\omega}}$, and hence, the feasible set $\{\mathbf{\boldsymbol{\omega}} \in F_{\mathbf{\boldsymbol{\omega}}} \colon \ \|B^{\top}L^{\dagger}\mathbf{\boldsymbol{\omega}}\|_{\infty} \leq \gamma_d \}$ renders a convex set. Notice that if $\omega_s \mathbf{1}_n \in F_{\mathbf{\boldsymbol{\omega}}}$, then $\mathbf{\boldsymbol{\omega}}^\star = \omega_s \mathbf{1}_n$ is the trivial optimal solution. Therefore, we assume tha the feasible design set does not contain $\mathbf{\boldsymbol{\omega}} = \omega_s \mathbf{1}_n$.

The problem of containing cascading failures in power systems can be stated as Problem \ref{prob:Optimal Budget Problem-Frequency design}, where remedial actions such as redispatch or load shedding (tuning $\mathbf{\boldsymbol{\omega}}$) are used to satisfy the constraints in \eqref{eq: Cohesiveness-Constrained Frequency Design}. In this case, an example for the cost function $g_{\mathcal{V}}(\mathbf{\boldsymbol{\omega}})$ is a penalty on the amount of load that has to be shed.

\ifx
\begin{remark}
	A closely related problem to cohesiveness-constrained frequency design is the so-called budget-constrained frequency design. In this problem, the goal is to minimize the phase cohesiveness given a certain tuning budget $C$ that the network designer can invest on tuning the natural frequencies.
\end{remark}

In the next section, we address the \emph{weight design problem} where the objective is to design the coupling strengths of a network in order to achieve a desired level of phase cohesiveness at a minimum design cost. \fi
\ifx
A special case of frequency design problem is uniform weights on the edges, i.e. $\mathbf{w}=k \mathbf{1}_n$. In this case, the phase cohesiveness is simply $\Phi(k,\boldsymbol{\omega})=k^{-1}\|B^\top (BB^\top )^{\dagger}\boldsymbol{\omega}\|_{\infty}$. The following problem addresses this case.
\begin{problem}[Joint Frequency-Weight Design for Homogeneous Coupling Weights] \label{prob:optimal critical coupling-Frequency}
	Given (\emph{i}) a graph with incidence matrix $B$ and uniform coupling strengths $\mathbf{w}=k \mathbf{1}_m$ (\emph{ii}) a frequency tuning function $g_{\mathcal{V}}\left(\boldsymbol{\omega}\right): \mathbb{R}^n \to \mathbb{R}^+$,	(\emph{iii}) a feasible design set $F_{\mathbf{\omega}}\subset\mathbb{R}^{n}$, (\emph{iv}) a desired level of phase cohesiveness ${\Phi}^{\star}\in\left[0,1\right)$, and (\emph{v}) a synchronized frequency $\omega_{s}$; find the optimal vector of natural frequencies, denoted by $\boldsymbol{\omega}^{\star}$, such that the network synchronizes at $\omega_{s}$ with cohesiveness level $\Phi^{\star}$ with minimum possible coupling.

	Problem \ref{prob:optimal critical coupling-Frequency} can be formulated as follows:
	\begin{equation}
	\begin{aligned}\underset{k> 0, \boldsymbol{\omega}\in F_{\mathbf{\boldsymbol{\omega}}}}{\min} & \quad k\\
	\mbox{s.t. } & \quad \|B^\top (BB^\top )^{\dagger}\boldsymbol{\omega}\|_{\infty} \leq k \Phi^{\star}\\
	& \quad g_{\mathcal{V}}\left(\boldsymbol{\omega}\right)\leq C,\\

	\end{aligned}
	\end{equation}
\end{problem}

\begin{rem}
	As was shown in section \ref{subsection: stability analysis}, at the critical coupling, the phase cohesiveness is equal to one. Hence, Problem \ref{prob:optimal critical coupling-Frequency} with $\Phi^{\star}=1$ finds the set of feasible natural frequencies that minimize the critical coupling.
\end{rem}
\fi

\section{\label{subsection: weight design}Weight design problem}

Consider a Kuramoto model $\mathcal{K}(B,\mathbf{w},\mathbf{\boldsymbol{\omega}})$ with a given network structure ($B$) and natural frequencies ($\mathbf{\boldsymbol{\omega}}$). In the weight design problem, our objective is to design the link weights ($\mathbf{w}$) to achieve a desired level of phase cohesiveness $\gamma_d \in [0, \pi/2)$ at a minimum design cost. In our setup, we assume that the network designer is able to tune the link weights within a convex feasible set $F_{\mathbf{w}} \subset \mathbb{R}^m_{+}$ that includes connected graphs. In other words, $\lambda_2(B \mbox{diag}(\mathbf{w})B^\top)>0$ for all $\mathbf{w}\in F_{\mathbf{w}}$. This spectral condition guarantees the connectivity of the resulting graph. Furthermore, the tuning cost is described by a function $f_{\mathcal{E}}(\mathbf{w})$, which we assume to be convex.

In most practical settings, however, the natural frequencies are subject to uncertainties. For example, in the context of power systems, the natural frequencies of the nodes correspond to net power generation (consumption) in generator (load) buses, and therefore, are subject to fluctuations depending on demand generation patterns \cite{FD-MC-FB:11v-pnas}.
Consequently, it is of practical relevance to extend the weight design problem in order to support uncertainties in the natural frequencies. In this direction, we assume that the natural frequencies of the oscillators belong to a convex uncertainty set, i.e., $\mathbf{\boldsymbol{\omega}} \in \Omega \subset \mathbb{R}^n$. Notice that the original weight design problem corresponds to the particular case in which the set $\Omega$ is a singleton. We formalize the (robust) weight design problem, as follows.

\begin{problem}\textbf{\emph{(Cohesiveness-constrained weight design)}} \label{problem: Budget-Constrained Weight Design}
	\label{prob:Optimal Cohesiveness Problem-Weight design} Assume we are given the following elements: (i) a connected undirected network with incidence matrix $B$, (ii) a compact convex set $\Omega \subset \mathbb{R}^n$ such that $\mathbf{\boldsymbol{\omega}} \in \Omega$, (iii) a convex weight-tuning cost function $f_{\mathcal{E}}\left(\mathbf{w}\right):\mathbb{R}_+^m \to \mathbb{R}$,
	(iv) a compact convex feasible design set $F_{\mathbf{w}}\subset\mathbb{R}_{+}^{m}$,
	and (v) a desired level of phase cohesiveness $\gamma_d \in [0,\pi/2)$. Find the optimal vector of link weights, denoted by $\mathbf{w}^{\star}$, that solves the following robust optimization problem,
	%
	\begin{align}\label{eq: robust weight design}
	\mathbf{w}^{\star} =\arg &\underset{\mathbf{w}\in F_{\mathbf{w}}}{\min} \ f_{\mathcal{E}}\left(\mathbf{w}\right) \\
	& \emph{s.t. } \underset{\mathbf{\boldsymbol{\omega}} \in \Omega}{\max} \ \|B^\top  ( B\emph{diag}({\mathbf{w}})B^\top )^{\dagger}\mathbf{\boldsymbol{\omega}} \|_{\infty}\leq \sin(\gamma_d). \nonumber
	\end{align}
	Then, by Criterion \ref{sync criterion}, the phase cohesiveness of the resulting network would satisfy $\max_{\mathbf{\boldsymbol{\omega}} \in \Omega} \varphi(B,\mathbf{w}^{\star},\mathbf{\boldsymbol{\omega}}) \leq \gamma_d$, and the minimum design cost is given by $f_{\mathcal{E}}(\mathbf{w}^{\star})$.
	\end{problem}

	In the context of power systems, Problem \ref{problem: Budget-Constrained Weight Design} corresponds to tuning the susceptance of the transmission lines within a feasible range such that the phase cohesiveness of the resulting network is bounded by $\gamma_d$, despite the fluctuations in power generations or consumption ($\mathbf{\boldsymbol{\omega}}$) across the generators and load buses, respectively.

	It can be verified that $\|B^\top ( B\mbox{diag}({\mathbf{w}})B^\top )^{\dagger}\mathbf{\boldsymbol{\omega}} \|_{\infty}$, appearing in the constraint of \eqref{eq: robust weight design}, remains unchanged by the substitution $\mathbf{\boldsymbol{\omega}} \leftarrow \mathbf{\boldsymbol{\omega}}-\frac{1}{n} \mathbf{1}_n \mathbf{1}_n^\top \mathbf{\boldsymbol{\omega}}$; hence, we can include the assumption $\mathbf{\boldsymbol{\omega}}^\top \mathbf{1}=\mathbf{0}_n$ in the uncertainty set $\Omega$.

	Notice that solving \eqref{eq: robust weight design} requires evaluating the term $\max_{\mathbf{\boldsymbol{\omega}} \in \Omega} \|B^\top (B^{\top}\mbox{diag}(\mathbf{w})B)^{\dagger} \mathbf{\boldsymbol{\omega}}\|$ for every feasible solution $\mathbf{w}$, which is not computationally tractable. We use robust optimization tools \cite{bertsimas2011theory} to convert this problem into a tractable form. The main idea behind this methodology is to use duality theory to replace the term $\max_{\mathbf{\boldsymbol{\omega}} \in \Omega} \|B^\top (B^{\top}\mbox{diag}(\mathbf{w})B)^{\dagger} \mathbf{\boldsymbol{\omega}}\|$ by its dual functions, i.e., a tight upper bound which does not involve the uncertain parameter $\mathbf{\boldsymbol{\omega}}$ and is characterized by the corresponding dual variables. Although this conversion can be done for any convex set $\Omega$, for simplicity in our exposition, we study the particular case of $\Omega$ being an interval uncertainty set, i.e.,
\begin{align} \label{eq: freqeuency_uncertainty_set}
\Omega=\{\mathbf{\boldsymbol{\omega}} \in \mathbb{R}^n \colon \ \underline{\mathbf{\boldsymbol{\omega}}} \leq \mathbf{\boldsymbol{\omega}} \leq \overline{\mathbf{\boldsymbol{\omega}}}, \ \mathbf{1}_n^\top \mathbf{\boldsymbol{\omega}} = 0\},
\end{align}
where $\underline{\mathbf{\boldsymbol{\omega}}}, \ \overline{\mathbf{\boldsymbol{\omega}}} \in \mathbb{R}^n$ are the vectors of lower and upper bounds on the natural frequencies, respectively. The next theorem provides a tractable formulation akin to \eqref{eq: robust weight design}.

\begin{theorem}\label{thm: robust weight design equivalent}
	Consider Problem \ref{problem: Budget-Constrained Weight Design} with $\Omega$ in \eqref{eq: freqeuency_uncertainty_set}. Let $\boldsymbol{b}_k \in \mathbb{R}^n$ be the $k-th$ column of the incidence matrix $B$. Then, the following optimization problem is equivalent to \eqref{eq: robust weight design},
	\begin{align} \label{eq: robust weight design equivalent}
	\text{\emph{min}}\quad &f_{\mathcal{E}}(\mathbf{w}) \\
	\mbox{\emph{s.t. }}\quad &\forall k \in [m], \nonumber \\

	&\overline{\boldsymbol{\lambda}}_k^\top \overline{\mathbf{\boldsymbol{\omega}}}-\underline{\boldsymbol{\lambda}}_k^\top \underline{\mathbf{\boldsymbol{\omega}}} \leq \sin(\gamma_d), \nonumber \\
	&\overline{\boldsymbol{\gamma}}_k^\top \overline{\mathbf{\boldsymbol{\omega}}}-\underline{\boldsymbol{\gamma}}_k^\top \underline{\mathbf{\boldsymbol{\omega}}} \leq \sin(\gamma_d), \nonumber \\
	&\underline{\boldsymbol{\lambda}}_k^\top -\overline{\boldsymbol{\lambda}}_k^\top +\nu_k\mathbf{1}_n^\top+\boldsymbol{b}_k^\top (B\mbox{\emph{diag}}({\mathbf{w}})B^\top )^{\dagger} =\mathbf{0}_n^{\top}, \nonumber \\
	&\underline{\boldsymbol{\gamma}}_k^\top -\overline{\boldsymbol{\gamma}}_k^\top +\eta_k\mathbf{1}_n^\top -\boldsymbol{b}_k^\top (B\mbox{\emph{diag}}({\mathbf{w}})B^\top )^{\dagger}=\mathbf{0}_n^{\top}, \nonumber \\
	&\overline{\boldsymbol{\lambda}}_k, \ \underline{\boldsymbol{\lambda}}_k, \  \overline{\boldsymbol{\gamma}}_k, \ \underline{\boldsymbol{\gamma}}_k \geq \mathbf{0}_n,\nonumber
	\end{align}
	with optimization variables $\mathbf{w}\in F_{\mathbf{w}}; \ \overline{\boldsymbol{\lambda}}_k,\underline{\boldsymbol{\lambda}}_k, \overline{\boldsymbol{\gamma}}_k, \underline{\boldsymbol{\gamma}}_k  \in \mathbb{R}^n_{+}; \nu_k, \eta_k \in \mathbb{R},\ k\in[m]$.
\end{theorem}
	\begin{pf}
		In Appendix \ref{appendix thm: robust weight design equivalent proof}.
	\end{pf}
Note that the feasible set of the weight design problem \eqref{eq: robust weight design} (or the particular formulation in \eqref{eq: robust weight design equivalent}) is not convex in the design variable $\mathbf{w}$, due to the pseudoinverse operation appearing in the constraints. In what follows, we propose a convex outer approximation to the feasible set of \eqref{eq: robust weight design equivalent} (in the space of $\mathbf{w}$) by virtue of the following lemma (which is a modification of a result in \cite{el1997cone}).
\begin{lemma} \label{lem: Schur}
	Consider a weighted, connected, undirected graph $G$, with the Laplacian matrix $L$. Define $\bar{L}^{\star}$ as the minimizer of the following semidefinite program,
	\begin{align} \label{eq: Schur complement}
		\bar{L}^{\star}= \arg & \ \underset{\bar{L}\in \mathbb{S}^{n\times n}}{\min} \quad \Tr(\bar{L}) \\
		\mbox{\emph{s.t. }} & \begin{bmatrix}
		L +\dfrac{1}{n}\mathbf{1}_n\mathbf{1}_n^\top  & I_n \\ I_n & \bar{L}\end{bmatrix} \succeq 0. \nonumber
	\end{align}
	Then, the Laplacian pseudo-inverse satisfies $L^{\dagger}=\bar{L}^{\star}-\frac{1}{n}\mathbf{1}_n\mathbf{1}_n^\top $.
\end{lemma}
\begin{pf}
		In Appendix \ref{appendix lem: Schur}.
\end{pf}
We now use Lemma \ref{lem: Schur} to propose a tractable convex relaxation of the optimization problem in \eqref{eq: robust weight design equivalent}. We regularize the objective function in \eqref{eq: robust weight design equivalent} with a penalty function $\Tr(\bar{L})$, and include the linear matrix inequality (LMI) in \eqref{eq: Schur complement} as an additional constraint in \eqref{eq: robust weight design equivalent}. After these modifications, we obtain the following convex outer approximation to \eqref{eq: robust weight design equivalent},
\begin{align}  \label{eq: robust weight design convex}
		\text{min}\quad & f_{\mathcal{E}}(\mathbf{w})+\alpha \Tr(\bar{L}) \\
		\mbox{s.t. }\quad &\forall k \in [m], \nonumber \\
		& \left[\begin{array}{cc}
		B\mbox{diag}(\mathbf{w})B^{\top}+\dfrac{1}{n}\mathbf{1}_n\mathbf{1}_n^\top & I_{n} \\
		I_{n} & \overline{L} \end{array}\right] \succeq0 , \nonumber \\

		& \overline{\boldsymbol{\lambda}}_k^\top \overline{\mathbf{\boldsymbol{\omega}}}-\underline{\boldsymbol{\lambda}}_k^\top \underline{\mathbf{\boldsymbol{\omega}}} \leq \sin(\gamma_d), \nonumber \\
		& \overline{\boldsymbol{\gamma}}_k^\top \overline{\mathbf{\boldsymbol{\omega}}}-\underline{\boldsymbol{\gamma}}_k^\top \underline{\mathbf{\boldsymbol{\omega}}} \leq \sin(\gamma_d), \nonumber \\
		& \underline{\boldsymbol{\lambda}}_k^\top -\overline{\boldsymbol{\lambda}}_k^\top +\nu_k \mathbf{1}^\top +\boldsymbol{b}_k^\top \overline{L}=\mathbf{0}_n^{\top}, \nonumber \\
		& \underline{\boldsymbol{\gamma}}_k^\top-\overline{\boldsymbol{\gamma}}_k^\top  +\eta_k \mathbf{1}^\top-\boldsymbol{b}_k^\top \overline{L} =\mathbf{0}_n^{\top}, \nonumber \\
		& \overline{\boldsymbol{\lambda}}_k, \ \underline{\boldsymbol{\lambda}}_k, \  \overline{\boldsymbol{\gamma}}_k, \ \underline{\boldsymbol{\gamma}}_k \geq \mathbf{0}_n, \nonumber
\end{align}
		where $\mathbf{w}\in F_{\mathbf{w}}$; $\bar{L} \in \mathbb{S}^{n\times n}$; $ \overline{\boldsymbol{\lambda}}_k,\underline{\boldsymbol{\lambda}}_k, \overline{\boldsymbol{\gamma}}_k, \underline{\boldsymbol{\gamma}}_k  \in \mathbb{R}^n_{+}$; $\nu_k, \eta_k \in \mathbb{R}$ are optimization variables, and $\alpha \in \mathbb{R}_{++}$ is a regularization constant.

\begin{remark}Notice that the regularization parameter $\alpha$ in \eqref{eq: robust weight design convex} should be large enough in order to make the LMI constraint in \eqref{eq: robust weight design convex} tight, i.e., the identity $\bar{L}^{\star}=(B\emph{diag}(\mathbf{w}^{\star})B^{\top}+\frac{1}{n}\mathbf{1}\mathbf{1}^{\top})^{-1}$ is enforced. On the other hand, too large values of $\alpha$ compromises the optimality of the objective function $f_{\mathcal{E}}(\mathbf{w})$. In practice, we can use a bisection search aiming to find the smallest value of $\alpha$ for which the LMI constraint is tight.
\end{remark}

\begin{remark}
In order for Lemma \ref{lem: Schur} to be applicable in \eqref{eq: robust weight design convex}, the feasible set $F_{\mathbf{w}}$ must be chosen such that $\lambda_2(B \emph{diag}(\mathbf{w})B^\top)>0$ for all $\mathbf{w}\in F_{\mathbf{w}}$. In some practical cases, we need to explicitly impose this constraint in our feasible design set (see, for example, Subsection \ref{subsection:Sparse Design}).
The following constraint can be included in the definition of $F_{\mathbf{w}}$ to guarantee a strictly positive algebraic connectivity \cite{boyd2006convex},
	\begin{align} \label{eq: connectivity constraint}
	B\emph{diag}(\mathbf{w})B^\top+ \dfrac{\beta}{n}\mathbf{1}_n\mathbf{1}_n^\top \succeq \beta I_n, \ 0<\beta\ll1.
	\end{align}
	To see this, notice that the eigenvalues of the left-hand side of \eqref{eq: connectivity constraint} is given by $\{\beta,\lambda_2,\ldots,\lambda_n\}$, where $\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $B\emph{diag}(\mathbf{w})B^\top$. The above LMI enforces that $\lambda_2(B\emph{diag}(\mathbf{w})B^\top) \geq \beta >0$; hence, the graph remains connected under \eqref{eq: connectivity constraint}. Furthermore, $\lambda_2(B\emph{diag}(\mathbf{w})B^\top)$ is directly connected to the speed of convergence towards the synchronous solution \cite{pecora1998master}. Therefore, we can also increase the convergence rate to the synchronous solution by increasing $\beta$.
\end{remark}

It can be verified that, if there is no uncertainty in the natural frequencies $\mathbf{\boldsymbol{\omega}}$ (i.e., if $\overline{\mathbf{\boldsymbol{\omega}}}=\underline{\mathbf{\boldsymbol{\omega}}}=\mathbf{\boldsymbol{\omega}}$), the convex relaxation of the weight design formulation in \eqref{eq: robust weight design convex} can be formulated as stated next.
\begin{corollary} Consider Problem \ref{problem: Budget-Constrained Weight Design} with $\Omega=\{\mathbf{\boldsymbol{\omega}}\}$ for a given $\mathbf{\boldsymbol{\omega}}\in \mathbb{R}^n$. The following optimization problem is a convex outer approximation to \eqref{eq: robust weight design},
		\begin{align} \label{eq: robust weight design convex no uncertainty}
		\underset{\mathbf{w}\in F_{\mathbf{w}},\overline{L}}{\mbox{\emph{minimize}}} &\ f_{\mathcal{E}}(\mathbf{w})+\alpha \Tr(\bar{L}) \\
		\mbox{\emph{subject to}} & \left[ \begin{array}{cc}
		B\mbox{\emph{diag}}(\mathbf{w})B^{\top}+\dfrac{1}{n}\mathbf{1}\mathbf{1}^{\top} & I_{n} \\
		I_{n} & \overline{L}\end{array} \right] \succeq0, \nonumber \\
		& \|B^\top \bar{L}\mathbf{\boldsymbol{\omega}}\|_{\infty}\leq \sin(\gamma_d). \nonumber
		\end{align}
\end{corollary}

In the next section, we provide numerical simulations to illustrate the effectiveness of the proposed framework in designing optimal networks of Kuramoto oscillators from the point of view of phase cohesiveness.

\section{Applications}
\label{section:NUMERICAL SIMULATIONS}

In the following subsections, we illustrate the use of our optimization framework in several problems of practical interest. Namely, we first consider the problem of finding the sparsest network able to achieve a desired level of phase cohesiveness. We then study the problem of designing the weights of the network when the natural frequencies are uncertain. Lastly, we use our framework to study the Braess' paradox\footnote{According to this paradox, an increase in the capacity of certain lines may result in a loss of synchrony.} in the context of power systems.

\ifx
\subsection{Power Redispatch}\label{subsection:Power Redispatchn}
In the context of power networks, redispatch refers to the process of tuning the generators' power injections across the network in order to relieve the stressing of transmission lines. Line overloads usually occur as a result of an imbalance in the distribution of power generations. In this situation, the redistribution of power injections is commonly used as a short-term remedial action to resolve these congestions.

The steady-state operating point of a lossless power network can be described by \eqref{eq: fixed point} where $\omega_i$ is the net power injected into node $i \in [n]$. The incidence matrix $B$ represents the connectivity of the network, and $w_e$ describes the susceptance of the transmission line $e \in [m]$.

PENDING...

	\begin{align*}
	\begin{aligned} &\underset{\boldsymbol{\Delta \omega}\in \mathbb{R}^n}{\min} \  \sum_{k \in \mathcal{V}_1} c_k|\Delta \omega_k|\\
	&\mbox{s.t. } \\
	& \ -\sin(\gamma_d) \leq \sum_{k \in \mathcal{V}_1} (L^{\dagger}_{i,k}-L^{\dagger}_{j,k})(\omega_{0,k}+\Delta \omega_k)\leq \sin(\gamma_d),\\
	& \ \sum_{k \in \mathcal{V}_1}  \Delta \omega_k=0, \\
	& \ \underline{\Delta \omega}_k \leq \Delta \omega_k \leq \overline{\Delta \omega}_k, \ k \in \mathcal{V}_1.
	\end{aligned}
	\end{align*}
\fi
\subsection{Sparsest network design}\label{subsection:Sparse Design}
Consider a Kuramoto model $\mathcal{K}(B_0,\mathbf{w}_0,\mathbf{\boldsymbol{\omega}}_0)$ where $B_0 \in \mathbb{R}^{n \times m_0}$ is the incidence matrix of the connection graph $G_0=(\mathcal{V},\mathcal{E}_0)$ with $n$ nodes and $m_0$ edges. The vectors $\mathbf{w}_0 \in \mathbb{R}^{m_{0}}_{+}$ and $\mathbf{\boldsymbol{\omega}}_0 \in \mathbb{R}^n$ contain the weights and natural frequencies of the Kuramoto network, respectively, which we assume to be  fixed and given. Our goal is to achieve a desired level of phase cohesiveness $\gamma_d \in [0,\pi/2)$ by adding as few new edges to the network as possible. We assume that we can only add new edges from a candidate set of $m_c$ edges $\mathcal{E}_c\subseteq \{\mathcal{V}\times \mathcal{V}\} \setminus \mathcal{E}_0$, i.e., a subset of edges not present in $G_0$, which we describe using the incidence matrix $B_c \in \mathbb{R}^{n \times m_c}$. As part of the design problem, not only we need to decide the location of the newly added edges, but also their weights, which we describe by the vector $\mathbf{w}_c \in F_{\mathbf{w_c}}\subset \mathbb{R}_{+}^{m_c}$. In this framework, the Laplacian of the upgraded network is given by $L=B_0\mbox{diag}(\mathbf{w}_0)B_0^\top+B_c\mbox{diag}(\mathbf{w}_c)B_c^\top$.

In accordance with the description above, our goal is to find the sparsest $\mathbf{w}_c$ such that the phase cohesiveness of the resulting network is at most $\gamma_d$. To
this end, we penalize the number of newly added edges using the $\ell_0$ norm of $\mathbf{w}_c$, $\|\mathbf{w}_c\|_0$. Therefore, the optimization problem to be solved is
\begin{align}\label{eq: l0_norm}
	\mathbf{w}_c^{\star} =\arg &\underset{\mathbf{w}_c\in F_{\mathbf{w}_c}}{\min} \|\mathbf{w}_c\|_{0} \\
	& \mbox{ \; s.t. } \; \; \varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}}_0) \leq \gamma_d, \nonumber
\end{align}
where $B$ is the incidence matrix of the set of \emph{selected} (nonzero) edges $\mathcal{E}=\mathcal{E}_0 \cup \{e \in \mathcal{E}_c  \colon w_e > 0\}$. The problem in (\ref{eq: l0_norm}) is combinatorial and, in general, hard to solve. A common convex relaxation for (\ref{eq: l0_norm}) is to substitute the $\ell_0$ norm by the $\ell_1$ norm $\|\mathbf{w}_c\|_1$. Upon this substitution, we can use Corollary \ref{problem: Budget-Constrained Weight Design} to solve the relaxed problem with $f_{\mathcal{E}}(\mathbf{w}_c)=\|\mathbf{w}_c\|_1$ as the cost function. However, using the  $\ell_1$ norm would result in an optimal network with a relatively large number of links with small weights. In order to promote sparsity, we propose to use re-weighted $\ell_1$ minimization algorithm in \cite{candes2008enhancing}. As the names implies, in this algorithm a sequence of weighted $\ell_1$ norm problems are solved, and in each round, the weights of the $\ell_1$ norm are updated in favor of promoting sparsity in the next round. The re-weighted $\ell_1$ minimization algorithm for solving \eqref{eq: l0_norm} is summarized in Algorithm \ref{tab:weight design for optimal cohesiveness}.
\ifx
\begin{align}\label{eq: l0_norm}
	\mathbf{w}_c^{\star} =\arg &\underset{\mathbf{w}_c\in F_{\mathbf{w}_c}}{\min} \|\mathbf{w}_c\|_{0} \\
	& \mbox{s.t. } \underset{ \{i,j\} \in \mathcal{E}}{\max} \lim\limits_{t \to \infty} |\theta_i(t)-\theta_j(t)| \leq \gamma_d, \nonumber
\end{align}
The constraint in \eqref{eq: l0_norm} imposes the phase cohesiveness requirement across the \emph{selected} (nonzero) edges, i.e., $\mathcal{E}=\mathcal{E}_0 \cup \{e \in \mathcal{E}_c  \colon w_e > 0\}$. The problem in (\ref{eq: l0_norm}) is combinatorial and, in general, hard to solve. A common convex relaxation of (\ref{eq: l0_norm}) is to substitute the $\ell_0$ norm by the $\ell_1$ norm $\|\mathbf{w}_c\|_1$. However, using the  $\ell_1$ norm would result in an optimal network with a relatively large number of links with small weights. In order to promote sparsity, we resort to re-weighted $\ell_1$ algorithm in \cite{candes2008enhancing}. As the name implies, the $\ell_1$ norm is replaced by \emph{weighted} $\ell_1$ norm, and in an iterative procedure, the weighting coefficients are updated in order to further enforce sparsity. The re-weighted $\ell_1$ scheme for solving \eqref{eq: l0_norm} is summarized in Algorithm \ref{tab:weight design for optimal cohesiveness}.
\fi

\begin{figure*}
\begin{center}
\includegraphics[width=1.01\textwidth]{Sparse_Network_Design_1}
\caption{Evolution of the sparsity pattern for the numerical example in Subsection \ref{subsection:Sparse Design}. In (a), we include the sparsity pattern of the adjacency matrix after the first iteration, which corresponds to a fully connected graph. Figures (b), (c) and (d) represent the networks obtained at iterations $3$, $7$, and $20$, respectively.}
									\label{fig: Sparse_Network_Design_1}
								\end{center}

							\end{figure*}

\begin{algorithm}
		{\footnotesize \vspace*{1ex}
			\textbf{Given}:\\
			$L_0=B_0 \mbox{diag}(\mathbf{w}_0)B_0^\top$, $\mathcal{E}_c$, $B_c=[\mathbf{b}_1,\cdots,\mathbf{b}_{m_c}]$, $\mathbf{\boldsymbol{\omega}}_0$, $F_{\mathbf{w}_c}$, $ \gamma_d$, $\alpha$, $k_{\max}$, and $0<\varepsilon\ll 1$.\\
			\begin{algorithmic}[1]
				\State set $\mathbf{p}^{(1)}=\mathbf{1}_{m_c}$ and $\widetilde{B}_c=B_c$; \label{step 1}
				\For {$k=1,\ldots,k_{\max}$}
				\State solve \eqref{eq: weighted L1 weight design} to obtain $\mathbf{w}_c^{(k)}$

						\begin{align} \label{eq: weighted L1 weight design}
		 					\mathbf{w}_c^{(k)}=& \arg \underset{\mathbf{w}_c\in F_{\mathbf{w}},\overline{L}}{\mbox{min}} \ \sum_{e=1}^{m_c} p_e^{(k)}|w_{c,e}|+\alpha \Tr(\bar{L}) \\
							\mbox{{s.t.}} & \left[ \begin{array}{cc}
							L_0+B_c\mbox{{diag}}(\mathbf{w}_c)B_c^{\top}+\dfrac{1}{n}\mathbf{1}\mathbf{1}^{\top} & I_{n} \\
							I_{n} & \overline{L}\end{array} \right] \succeq0, \nonumber \\
							& \|[B_0 \ \widetilde{B}_c]^\top \overline{L}\mathbf{\boldsymbol{\omega}}_0\|_{\infty}\leq \sin(\gamma_d), \nonumber
													\end{align}
				\State update $p_{e}^{(k+1)}=(\varepsilon+w_{c,e}^{(k)})^{-1},\ e \in [m_c]$;
				\State update $\widetilde{B}_c=\left[\mathbf{b}_{e}\right]_{\{e\colon w_{c,e}^{(k)}>\varepsilon\}}$;

				\EndFor
			\end{algorithmic}}
			\caption{\hspace*{-.5ex}: sparse weight design} \label{tab:weight design for optimal cohesiveness}
		\end{algorithm}

							\begin{figure}
								\begin{center}
										\includegraphics[width=0.4\textwidth]{Sparse_Network_Design_2.pdf}
										\caption{Time evolution of phases  for the numerical example in Subsection \ref{subsection:Sparse Design}. The phase cohesivness satisfies the design requirement, i.e., $\varphi(B,\mathbf{w}^\star,\mathbf{\boldsymbol{\omega}}_0)=1.3119<1.413$.}
										\label{fig:Sparse_Network_Design_2}
								\end{center}
							\end{figure}

		\emph{Description of the algorithm}: In Step 1, the coefficients of the $\ell_1$ norm are initialized at one (i.e., $\mathbf{p}^{(1)}=\mathbf{1}_{m_c}$), and the incidence matrix of the selected edges ($\widetilde{B}_c$) includes all the candidate edges $\mathcal{E}_c$. In Step 3 of iteration $k$, the optimization problem \eqref{eq: weighted L1 weight design} is solved to obtain the optimal edge weights $\mathbf{w}_c^{(k)}$. In Step 4, the components of $\mathbf{p}^{(k)}$ are updated inversely proportional to the corresponding components of $\mathbf{w}_c^{(k)}$. The constant $0<\varepsilon \ll 1$ is used to avoid singularity. In Step 5, $\widetilde{B}_c$ is updated to include the selected (nonzero) edges from Step 3. Steps 2 to 6 are repeated for a specified number of iterations, denoted by $k_{\max}$.

	In our numerical experiments, we assume that $n=30$, $L_0=\mathbf{0}_{n\times n}$,  $B_c=B_{K_n}$, $\omega_{0,i}=-1+2\frac{i-1}{n-1}$ for $\ i\in [n]$, $F_{\mathbf{w}}=[0,10]^{m_c}$, and $\gamma_d=1.413$ rad (so that $\sin(\gamma_d)=0.99$), $\alpha=2$, $k_{\max}=10$, and $\varepsilon=0.01$. In other words, the initial network is empty and the network designer is allowed to connect any pair of nodes to construct the sparsest synchronizable network (since $\sin(\gamma_d)\lesssim 1$). To maintain the connectivity of the network, we include the LMI \eqref{eq: connectivity constraint} with $\beta=10^{-4}$ in the definition of $F_{\mathbf{w}}$. Fig. \ref{fig: Sparse_Network_Design_1} illustrates the evolution of the sparsity pattern of the adjacency matrix as Algorithm \ref{tab:weight design for optimal cohesiveness} progresses. The first iteration of the algorithm, in Fig. \ref{fig: Sparse_Network_Design_1}-(a), is a fully connected graph, while the sparsest graph in the final iteration, in Fig. \ref{fig: Sparse_Network_Design_1}-(d), contains only $8\%$ of the edges. Fig. \ref{fig:Sparse_Network_Design_2} illustrates the time evolution of the phases for the designed sparse network, in Fig. \ref{fig: Sparse_Network_Design_1}-(d). The resulting phase cohesiveness satisfies the design constraint $\sin(1.0378)=0.8613
	<0.99$ as expected.

\subsection{Robust synchronization of distributed analog clocks} \label{example: Critical coupling weights}
\ifx In this section, we study the problem of distributed synchronization of coupled analog clocks. \fi Consider a wireless sensor network consisting of $n$ processors $\mathcal{V}=\{1,\cdots,n\}$ equipped with analog clocks. In order to efficiently perform distributed computations across the network, the clocks are required to synchronize their phases. In this context, the Kuramoto model can be used as a distributed synchronization scheme for synchronizing the phases \cite{simeone2008distributed}. Therefore, we can use \eqref{eq: Kuramoto model-matrix} as a model of analog clock synchronization, where $B \in \mathbb{R}^{n\times m}$ is the incidence matrix of the communication graph, $\mathbf{w} \in \mathbb{R}^{m}_{+}$ is the vector of connection strengths, and $\mathbf{\boldsymbol{\omega}} \in \mathbb{R}^n$ is the vector of natural frequencies of the clocks. The incidence matrix $B$ is determined by the geographical distribution of the processors and their communication range. In practice, the natural frequencies $\omega_i \ i\in[n]$, are uncertain due to hardware imperfections and aging.
\begin{figure*}
	\centering
	\includegraphics[width=1.0\textwidth]{Robust_Network_Design.pdf}
	\caption{The sensor network of subsection \ref{example: Critical coupling weights}. (a) the initial structure, (b) the designed structure, and (c) the time evolution of phases for the designed network for the worst case realization of the natural frequencies.}
	\label{fig: sensor network}
\end{figure*}
Therefore, the communication graph must be designed in order to synchronize the clocks in the presence of uncertainties in the natural frequencies. We consider the network design problem in \eqref{eq: robust weight design} with
\ifx
\begin{align}
	\mathbf{w}^\star(\gamma_d) =\arg &\underset{\mathbf{w}\in F_{\mathbf{w}}}{\min} \|\mathbf{w}\|_{1} \\
	& \mbox{\:  s.t.  \:} \max_{\mathbf{\boldsymbol{\omega}} \in \Omega}\varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}}) \leq \gamma_d, \nonumber
\end{align}
\fi
sum of the weights $\sum_{e=1}^{m} w_e=\|\mathbf{w}\|_1$ as the cost function. For numerical simulations, we consider the sensor network depicted in Fig. \ref{fig: sensor network}-(a)  with $n=30$ processors and $m=56$ links. We assume that the natural frequencies are nominally equal to $1$ with $20\%$ uncertainty, i.e., in \eqref{eq: freqeuency_uncertainty_set} we have $\underline{\mathbf{\boldsymbol{\omega}}}=-0.2\,  \mathbf{1}_n$ and $\overline{\mathbf{\boldsymbol{\omega}}}=+0.2\, \mathbf{1}_n$. We then solve \eqref{eq: robust weight design convex} with $\gamma_d=\pi/10 \ \mbox{rad}$, and the feasible design set being the positive orthant, $F_{\mathbf{w}}=\mathbb{R}_{+}^{m}$.
The resulting network is illustrated in Fig. \ref{fig: sensor network}-(b), where the optimal cost is $\|\mathbf{w}^\star\|_1=70.01$. We then numerically simulate the phase dynamics of the designed network using the \emph{worst-case} vector of natural frequencies $\widetilde{\mathbf{\boldsymbol{\omega}}}$, where
$$\widetilde{\mathbf{\boldsymbol{\omega}}}=\arg \max_{\mathbf{\boldsymbol{\omega}} \in \Omega} \|B^\top (B\mbox{diag}(\mathbf{w^{\star}})B^\top)^{\dagger} \mathbf{\boldsymbol{\omega}}\|_{\infty}.$$
The resulting evolution is plotted in Fig. \ref{fig: sensor network}-(c). We observe that $\lim_{t\to\infty}\|B^\top \theta(t)\|_{\infty}=\pi/10$ as expected; hence, the phase cohesiveness of the optimal network is guaranteed to be less that $\pi/10$ for all $\mathbf{\boldsymbol{\omega}} \in \Omega$.

\subsection{Braess' paradox in power systems} \label{example: Braess's Paradox}
In the context of network synchronization, the Braess' paradox describes the counter-intuitive phenomenon that adding new links (or strengthening existing ones) may lead to loss of synchrony. To illustrate this fact, let us consider a lossless power network \footnote{For a lossless power network, the steady state operating point of the network is described by \eqref{eq: fixed point}. Therefore, the results in this paper are applicable to lossless power networks.} represented in Fig. \ref{fig:Braess_Paradox_1}-(a) (originally proposed in \cite{witthaut2012braess}), which we will refer to as $G_0$. This network has $4$ generators (orange nodes), $4$ load buses (green nodes), and $m_0=10$ transmission lines (solid lines) . All nodes are assumed to have the same value of power demand/generation, in particular,  $\omega_{i}=0.95$ for generators and $\omega_{i}=-0.95$ for load buses. Furthermore, all the edges in $G_0$ are assumed to have identical capacity equal to $1$. For these numerical values, the phase cohesiveness satisfies $\sin(\varphi(B_0,\mathbf{w}_0,\mathbf{\boldsymbol{\omega}}))=0.95$, where $B_0$ is the incidence matrix of $G_0$, and $\mathbf{w}_0=\mathbf{1}_{10}$.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.3\textwidth]{Braess_Paradox_1_1.pdf}
		\caption{Power network of subsection \ref{example: Braess's Paradox}. The load buses are denoted by green nodes and the generators are denoted by red nodes. In (a) the dashed lines are the set of potential new edges to be added to the network, and the solid lines are the existing lines. In (b) the red lines denotes the added lines after the optimization.}
		\label{fig:Braess_Paradox_1}
	\end{center}
\end{figure}

We consider the problem of adding new lines to the network (chosen from a set of candidate lines) in order to decrease the phase cohesiveness below $\gamma_d=\pi/3$. Notice that thermal limit constraints on the transmission lines are precisely equivalent to bounds on phase cohesiveness \cite{FD-MC-FB:11v-pnas}. The candidate lines are indicated by dashed lines in Fig. \ref{fig:Braess_Paradox_1}-(a). We denote the subgraph induced by the candidate lines as $G_c$, their incidence matrix as $B_c \in \mathbb{R}^{8 \times 7}$, and their weights as $\mathbf{w}_c\in\mathbb{R}_{+}^7$.

We would like to minimize the total capacity (measured as the $\ell_1$ norm of $\mathbf{w}_c$) added to the network. Hence, the optimization problem to be solved is
\begin{align} \label{eq: Braess_Paradox_Optimization}
	\mathbf{w}_c^\star(\gamma_d) =\arg &\underset{\mathbf{w}_c\in F_{\mathbf{w}_c}}{\min} \|\mathbf{w}_c\|_{1} \\
	& \mbox{\; \,  s.t.  \;\,} \varphi(B,\mathbf{w},\mathbf{\boldsymbol{\omega}}) \leq \gamma_d, \nonumber
\end{align}
where $B=[B_0 \ B_c]$ and $\mathbf{w}=[\mathbf{w}_0^{\top} \ \mathbf{w}_c^{\top} ]^{\top}$ are, respectively, the incidence matrix and weight vector of the augmented network. We then solve \eqref{eq: robust weight design convex no uncertainty} for $f_{\mathcal{E}}(\cdot)=\|\cdot\|_1$, and $F_{\mathbf{w}_c}=\mathbb{R}^{7}_{+}$.  The resulting network is depicted in Fig \ref{fig:Braess_Paradox_1}-(b). The selected (nonzero) edges are $w_{28}$, $w_{38}$, and $w_{26}$, and the remaining candidate links ($w_{34}$, $w_{24}$, $w_{16}$, and $w_{18}$) are set to zero.

To relate the result of our optimization to the Braess' paradox, we run the following experiment: increase the capacity of $w_{34}$ from $1$ to $2.2$ and plot the variation of $\|B_0^\top L^\dagger \mathbf{\boldsymbol{\omega}} \|_{\infty}$ as a function of $w_{34}$. This variation is plotted in Fig \ref{fig:Braess_Paradox_2}-(a) and observe how, as we increase the link strength $w_{34}$, the value of $\|B_0^\top L^\dagger \mathbf{\boldsymbol{\omega}} \|_{\infty}$ increases monotonically, and crosses the stability threshold at $w_{34}\approx 1.62$. In other words, increasing the value of $w_{34}$ has a detrimental effect on the network stability. To validate our claims, we plot the time evolution of the phase dynamics for $w_{34}=1<1.62$ (Fig. \ref{fig:Braess_Paradox_2}-(b)) and $w_{34}=2>1.62$ (Fig. \ref{fig:Braess_Paradox_2}-(c)), in which we observe how the network dynamics transition from a stable to an unstable regime. Similar results can be obtained when we increase $w_{16}$.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.45\textwidth]{Braess_Paradox_2.pdf}
		\caption{(a) Variation of $\|B^\top L^{\dagger} \mathbf{\boldsymbol{\omega}} \|_{\infty}$ as a function of $w_{34}$. (b) Time evolution of phases when $w_{34}<1.62$. (c) Time evolution of phases when $w_{34}>1.62$.}
		\label{fig:Braess_Paradox_2}
	\end{center}
\end{figure}

Similarly, we consider the effect of adding a new line connecting nodes 2 and 4 by increasing its capacity from $0$ to $1$. In Fig. \ref{fig:Braess_Paradox_3}, we plot the variation of $\|B_0^\top L^\dagger \mathbf{\boldsymbol{\omega}} \|_{\infty}$ as we increase  $w_{24}$, and observe that when $w_{24}\approx 0.41$ the network transition from a stable to an unstable state. We validate this claim by plotting the time evolution of the phase dynamics for $w_{24}=0<0.41$ (Fig. \ref{fig:Braess_Paradox_3}-(b)) and $w_{34}=0.8>0.41$ (Fig. \ref{fig:Braess_Paradox_3}-(c)).  Similar results can be obtained when we increase $w_{28}$.

The above observations confirm that the optimization problem \eqref{eq: Braess_Paradox_Optimization} has assigned zero weight to those links that are detrimental for phase cohesiveness. More generally, our optimization framework which is based on Criterion \ref{sync criterion} is capable of identifying those lines that give rise to Braess' paradox.

\begin{figure}
	\begin{center}
			\includegraphics[width=0.47\textwidth]{Braess_Paradox_3.pdf}
			\caption{(a) Variation of $\|B^\top L^{\dagger} \mathbf{\boldsymbol{\omega}} \|_{\infty}$ as a function of $w_{24}$. (b) Time evolution of phases when $w_{24}<0.41$. (c) Time evolution of phases when $w_{24}>0.41$.}
			\label{fig:Braess_Paradox_3}
	\end{center}
\end{figure}

\section{Conclusions}
\label{section:CONCLUSIONS}
In this paper, we have proposed a convex optimization framework for designing the natural frequencies and the coupling weights in a network of nonidentical Kuramoto oscillators. We have used phase cohesiveness as our design constraint, as it captures both the steady-state performance and stability of the network.
In this context, we have addressed the following network design problems:
(\emph{i}) the nodal-frequency design problem, in which we design the natural frequencies of the oscillators for a given network, and
(\emph{ii}) the edge-weight design problem, in which we design the edge weights.
For the latter case, we have also developed a robust framework to design networks when the natural frequencies are uncertain. We have illustrated the applicability of our results using several network design problems of practical interest, namely, a sparsity-promoting design problem (Subsection \ref{subsection:Sparse Design}), the design of a network of distributed wireless analog clocks in the presence of uncertainties (Subsection \ref{example: Critical coupling weights}), and a power network design problem in which we illustrate how our framework is able to detect links whose presence are detrimental for synchronization (i.e., the Braess' paradox), as illustrated in Subsection \ref{example: Braess's Paradox}.

redpajama_set_name: RedPajamaArXiv


It was nice and peaceful on New Year's Day in Osaka. We got in from Hiroshima just in time to ring in the new year quitely hanging out and eating at a restaurant the night before.
New Year's in Japan is quiet unlike the fire work shows in the neighborhoods of Hawaii. Japan's shrines and temples are open for people to go and visit from midnight and well into the day. We spent some time to get some desserts at a cafe before we took some time to go into the city to see the festivities.
Thanks again for reading. Take care.


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text:
How have Cardiff City's last five managers fared since leaving the Bluebirds?
It's safe to say they've had mixed success since leaving the Welsh capital
Sign up to FREE daily email alerts from Wales Online - Weekday Cardiff City FC
Neil Warnock has worked wonders since his arrival at Cardiff City having steered the Bluebirds away from the Championship's relegation zone following the club's disastrous start to the season under Paul Trollope.
The Yorkshireman has an impressive 41% win rate with the Bluebirds having collected 14 victories in his 34 games in charge of the club.
What we learned about the talent coming through at Cardiff City as the under-23s took on Bristol City
Warnock has helped to galvanise the whole club and is hoping to lead the Bluebirds into the promised land of the Premier League next season.
Cardiff Manager Neil Warnock thanks the fans at full time. (Image: Huw Evans Picture Agency)
But what has become of the men who occupied the Cardiff City hotseat before him?
Here we take a look and how the Bluebirds' last five permanent managers have fared and deliver our verdict as to whether or not they've improved since leaving Wales...
Cardiff City u23s v Bristol City u23s RECAP: Youthful Bluebrids development side beaten 2-0
Paul Trollope
Cardiff record: P12; W2; D2; L8
It proved to be a truly disastrous reign for Paul Trollope who secured just two wins from his 11 league matches in-charge of the Bluebirds.
He also oversaw his side crash out of the League Cup to his former club and Cardiff's Severnside rivals Bristol Rovers in August.
The Bluebirds were left languishing in the Championship's relegation zone when the club's hierarchy made the swift decision to axe Trollope, with Warnock arriving just days after the former Wales international's sacking.
Brighton assistant manager and former Cardiff City man Paul Trollope
But the 44-year-old won't be bumping into his former employers any time soon having just helped Brighton secure promotion into the Premier League.
Trollope has been Chris Hughton's assistant manager at the AMEX Stadium since November and could see the Seagulls secure the Championship title against Bristol City this weekend.
Verdict: It was a truly torrid time for Trollope with the Bluebirds, but having secured promotion with the Seagulls, it's safe to say the move has been a good one for the ex-Cardiff boss.
Former Cardiff City manager Dave Jones axed from his job amid raging war of words with Sky Sports presenter
Russell Slade
Cardiff record: P86; W32; D27; L27
Having seen the Bluebirds relegated from the Premier League just months earlier, Russell Slade was the man tasked with stabilising the club and reducing Cardiff's wage bill following the departure of Manchester United legend Ole Gunnar Solskjaer.
After a turbulent debut campaign in which the club finished 11th, Slade enjoyed a significantly more settled second season as the Bluebirds challenged for a play-off place.
But a run of just one win in their last seven league games saw the Bluebirds finish in eighth place, six points adrift of a top six spot.
Former Cardiff City manager Russell Slade (Image: Getty Images)
It led to Slade being moved from manager to Cardiff's head of football, although the switch did not last as he resigned from the role less than a month later.
And in the year since his departure from the Welsh capital, Slade has certainly fallen into the bracket of having more clubs in his career than Tiger Woods.
Ill-fated stints with Charlton Athletic and Coventry City did little to enhance his reputation in the Football League.
And Slade was recently appointed as manager of League Two outfit Grimsby Town for the second time in his career, with the Mariners winning two of their three games so far under the leadership of Slade.
Verdict: Many Bluebirds feel that Slade is now at his level with Grimsby, and following his disastrous spells with Coventry and Charlton, it's clear that his managerial career has been on a significant downward spiral since departing Cardiff less than a year ago.
How big does Cardiff City's squad need to be to challenge for promotion into the Premier League next season?
Cardiff record: P30; W9; D5; L16
One of only two men to manage the Bluebirds in the Premier League, Solskjaer enjoyed the perfect start to life with Cardiff as his side came from behind to beat Newcastle 2-1 in the third round of the FA Cup.
But the Bluebirds lost their opening three league games under the Norwegian, two of which came against Manchester United and Man City.
And Cardiff won just three times in 18 Premier League games under Solskjaer's watch as they made a swift return to England's second tier, with the Norwegian being sacked just a month into the 2014-15 Championship season.
Ole Gunnar Solskjaer when he was appointed as manager of Cradiff City
After more than a year out of the game, Solskjaer returned to former club Molde - where he started his first team managerial career and also spent a season during his playing days - in October 2015.
He boasts an impressive 52% win rate over the 44 games he has managed in his second stint with Molde who are currently fifth in Eliteserien after five games of the new season.
Verdict: To go from the Premier League to a significantly weaker league in his native land is no doubt a drop in managerial standards, although the Norwegian is still learning his trade and could benefit in the long term from being kept out of the limelight in the early days of his career in the dugout.
Neil Warnock will target more attacking options in the summer after admitting Cardiff City were blunt against Wigan Athletic
Malky Mackay
Cardiff record: P125; W54; D37; L34
The Scot boasts the highest win percentage (43.2%) of the men on this list and will always be highly regarded in the Welsh capital after leading the Bluebirds to the Championship title in the 2012-13 season.
But after falling out with owner Vincent Tan, Mackey was sacked by the Bluebirds midway through their debut Premier League campaign.
He later became embroiled in a row over a series of text messages sent during his time as Cardiff manager in which he was accused of sending racist content, an accusation he strongly denied.
Former Cardiff City boss Malky Mackay (Image: Getty Images)
Back on the pitch, Mackay became the unpopular choice as Uwe Rosler's replacement at Wigan, although he was dismissed after just 25 games in-charge of the Latics where he secured only five victories.
He is currently working as the Scottish Football Association's performance director, being appointed just weeks after Scotland's 3-0 World Cup qualifying defeat to England at Wembley in November.
Verdict: Off-the-field problems have blighted Mackay's career since he was axed by the Bluebirds in December 2013, and given that he's no longer in management after a woeful spell with Wigan, it's fair to say the Scot has gone backwards in his managerial career in recent years.
Cardiff City must search for star quality in the summer as Wigan Athletic stalemate shows how difficult promotion will be - The final word
Cardiff record: P315; W132; D84; L99
Close but no cigar.
The perfect phrase to describe Dave Jones' time in the Welsh capital.
Jones famously led the Bluebirds to the FA Cup final in 2008 and oversaw the club secure three consecutive top seven finishes in his final three seasons with the club, with the final two being fourth-placed finishes.
A heart-breaking play-off final defeat to Blackpool at Wembley in 2010 was followed by a semi-final play-off defeat to Reading a year later, a defeat that cost Jones his job.
Dave Jones on the touchline at Wembley alongside Harry Redknapp
After 10 months away from the game, Jones was named as Sheffield Wednesday manager in March 2012 following the controversial sacking of Gary Megson who left the club in third place in League One.
The Owls went on to secure promotion under Jones and finished in 18th in their first season back in the Championship.
But a disastrous start to his second full season in-charge saw him sacked after securing just one win in his opening 16 league games in the 2013-14 campaign.
He was out of management for just over three years before being appointed as manager of League Two strugglers Hartlepool in January this year.
And it's safe to say he endured something of a nightmare at Victoria Park, winning just four of his 18 games as Pools boss before being sacked following Sky Sports presenter Jeff Stelling's furious on-air rant calling for the former Cardiff manager to be axed .
Verdict: His time with Hartlepool was an unmitigated disaster to say the least, and he's left the club on the brink of exiting the Football League following Newport County's sensational revival under caretaker boss Mike Flynn. Those days of challenging for a Premier League spot will seem like a distant memory for Jones.
Neil Warnock gives his verdict on Mark Harris and Ibrahim Meite as Bluebirds rookies make league debuts
ShoppingSony's PlayStation 5 could be on sale within days, according to latest leakAn anonymous source posting on 4chan claims the new gaming console will be unveiled within days
UK NewsMan labelled 'a monster' for unusual way of transporting books'My colleague called me a 'book murderer''

redpajama_set_name: RedPajamaCommonCrawl


text:
Notes of interest from Apple's Q3 2010 conference call
AppleInsider Staff | Jul 20, 2010
Apple reported yet another impressive quarter Tuesday, with record sales of 3.47 million Macs helping to propel the company to $3.25 billion in profits, a 77 percent increase from a year prior. Following the news, Apple executives conducted a financial conference call with analysts and the press, and notes of interest follow.
The third quarter of Apple's 2010 fiscal year produced $3.5, and the company posted revenue of $15.7 billion. The company sold 8.4 million iPhones, 9.41 million iPods, and 3.27 million iPads.
On Tuesday's call were Apple Chief Financial Officer Peter Oppenheimer and Chief Operating Officer Tim Cook.
Apple's regional business segments
Apple ended the quarter with $45.8B in cash, up from $41.7B at end of last quarter.
Mac sales were up 40 percent in the Americas, with revenue up 25 percent.
Apple's biggest overseas growth came in Asia Pacific, where Mac unit sales were up 71 percent year over year. Revenue in that market was up a whopping 160 percent from Mac sales.
Worldwide, portables increased 37 percent, while desktops were down 12 percent. Total Mac sales were up 18 percent.
"We have a lot of opportunity domestically and in other parts of the Americas," Cook said of iPhone sales. But they're growing faster internationally for all products, when compared to domestically.
"The company's revenues in the Americas are growing over 40 percent," he said. "And so this is a huge number. It's not that we're talking about a low number, it's just that the international numbers are absolutely killer."
Apple's Mac business
3.47 million Macs, exceeding record by over 100,000. Represents 33 percent year over year growth.
Double-digit Mac growth in each geography. No supply issues in the last quarter.
"There's clearly a move to mobility," Cook said. Desktop numbers will grow when there's a new machine, but in quarters like the last, the desktops continue to lose share.
Apple's iPhone business
iPhone had an average selling price of $595.
Over 5 billion apps downloaded from the App Store.
Strong growth in Asia, Europe and Japan.
Surpassed sales of 100 million iOS devices in the quarter
Will defer about $175 million to buy cases. Cost will be recognized upon delivery to customers.
400 higher education institutions have approved the iPhone for faculty, staff and students. Over 60 percent of Fortune 500 are deploying the iPhone.
iPhone demand was ramping in June. "We are still ramping and increasing volume, but again there's not a specific thing... it's just a matter of getting up the ramp," Cook said.
Greatly reduced shipments of previous-generation iPhone around June 7. New iPhone 4 and 8GB iPhone 3GS didn't launch until three days before the end of the quarter. "We had significantly lower sales after June 7, for that period of time as we waited on the official launch," Cook said.
Growth rate prior to June 7 was around 90 percent. Exited the quarter with total iPhone sales up 61 percent.
Cook said the antenna issue has had no impact on sales. "We are selling every unit we can make currently." He said demand has been "absolutely stunning."
Returns for iPhone 4 are less than iPhone 3GS. "The ones for this specific (antenna) issue are extremely small," Cook said.
iAds: "We're going to learn a lot this calendar year and build a foundation for the future. And beyond that, I don't have any further specifics to share with you today," Oppenheimer.
North Carolina data center is "on schedule and everything is going fine." Expect to be completed and be in use by the end of the calendar year, Oppeneheimer said.
Bumpers: They will need to defer revenue for the iPhone 4s they sell where they have not delivered the bumpers, and where they have not heard from the customers wanting to place an order. Expect to make a revenue accrual with no cost of about $175 million. Expect to have it recorded as revenue in the September quarter. They will expense the cost of the bumpers when they ship them to customers. "Our most important objective there is to take care of every customer," Oppenheimer said.
Spain will go from one exclusive carrier to three carriers at the end of the month. They are still dealing with exclusivity on a case-by-case basis. "I just see an enormous amount of opportunity there," Cook said.
Possibility of other carriers: "We are very happy to be a partner with AT&T," Cook said. They have been a first-class partner and have really pioneered the smartphone growth."
Apple's iPad business
$2.17 billion sale of iPad and accessories. iPad sales alone was about $2.1 billion.
Average selling price of iPad was about $640.
iPad launches in 9 additional countries on July 23.
Already have 50 percent of the Fortune 100 deploying or testing the iPad. "It's incredible," Cook said.
Backlog of demand at end of last quarter, unable to fill. Selling them as fast as they can. "We are still are quoting longer lead times than we'd like, and we're working around the block to get supply and demand in balance," Cook said. "In the scheme of things, it's a good problem to have."'
Apple is working on increasing capacity, and feels like they can. They were just caught off guard by the demand.
When will supply for the iPad meet demand? "We honestly don't know," Cook said. The company was honestly surprised by its initial sales.
iPad is not taking a typical early adopter curve. But Apple believes the market is "very big," and that iPad is "defining the market." "We want to take full advantage of it, and so we are investing enormous time and resources in increasing our capability in getting iPad out to as many people as we can," Cook said.
"The demand has just been amazing and this is off to an incredible start," Oppeneheimer said.
On potential cannibalization of the Mac or other products: "We've only been selling for three months, and so I think the real answer is it's too early to tell," Cook said. He noted that the Mac had its best quarter ever when the iPad sold over 3 million units. "For us it's a jaw dropper."
Will iPad margins increase over time? Oppenheimer said they were "purposefully aggressive" when they launched the iPad. They are always working to become more efficient and cut down costs, and he expects it to continue with other products.
"We are absolutely selling every unit that we can make," Cook said. He said sales have been positive in every country they've released it in thus far. Anecdotally, he believes it's already beyond an "early adopter" phase. "I think it's extremely unique and extremely successful."
Possibility of an iPad halo effect, selling more Macs and other products? "We'll see. I don't want to predict it," Cook said. Said there's still a great opportunity for the Mac share to grow, because it is still so small.
"This is where it's great to have a lower share, because if it turns out the iPad cannibalizes PCs, I think it's fantastic for us," Cook said. "Because there's a lot of PCs to cannibalize. It's still a big market."
Apple's iPod business
Sales of iPod touch are very strong, sales are up 48 percent year over year.
ASP for iPod touch increased 12 percent, and revenue grew 4 percent. Overall iPod ASP dropped about $7.
iPod market share is over 70 percent. Also the top-selling player internationally.
Apple's retail business
Retail revenue up 73 percent. Mac sales were up 38 percent.
About half of Macs were sold to customers who never owned a Mac before.
7 new stores opened in the quarter, including 4 in U.S., 2 in Australia and 1 in Canada, ending with 293. With an average of 280 stores open, average revenue per store was $9 million.
Expect to open 24 new stores this quarter, including Paris and Shanghai. Will also have first two stores in Spain in Barcelona and Madrid.
Stores catered to 60.5M visitors, compared to 38.6M during year ago.
Apple's next (Q4 2010) fiscal quarter
EPS of $3.44 for the fourth fiscal quarter of 2010. Forecast 35 percent gross margin.
Higher mix of iPhone 4 and iPad sales expected, a negative for gross margin.
Oppenheimer said that some adjustments with suppliers occurred in June. Gross margin for the September quarter, they are delivering "great value" to customers, and the products have a "higher cost structure."
"We have been pretty aggressive here with pricing, and it's going to play true of it on the margin line," Oppenheimer said.
Apple plans to defer the revenue related to free iPhone bumpers till the end of the quarter. Estimated to cost $175M.
I expect Apple to put a camera in all iPods coming September.
joe hs
Originally Posted by ihxo
even the shuffle
I'm not too sure about the classic either; but the touch gaining a camera (or two) should be apples next move.
skiracer1987
Considering the economy we are in this is incredible. Thing is, I can only see further upside to all of this.
Apple is definitely doomed
sacto joe
I agree. And within six months (just in time for xmas?) there'll be FaceTime across all Apple devices with a camera. Next iPad upgrade, that'll be there as well.
There's a term for this: Synergy. Apple's tapped into the energy of synergy.
The future's so bright, we gotta wear shades!
jpcg
This is incredible!
Also just got an Email that my iPhone 4 just shipped, 2 day early!
edit: I waited 3 Weeks to get to this Point
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